Double integrals over rectangles — Fubini's theorem

4.4.16 · Maths › Multivariable Calculus

Double integral HAI kya?

YEH definition kyun? Har term $f (x_{i}^{*}, y_{j}^{*}) Δ A$ ek patli rectangular column ka volume hai: base $Δ A$ , height $f$ . Saari columns jodo → surface ke neeche ka approximate volume. Base ko shrink karo → exact volume. Yeh bilkul waisi hi idea hai jaise 1-D area, bas ek dimension upar.

Hum actually compute kaise karte hain? — Iterated integrals

2-D limit directly compute karna bahut mushkil hai. Fubini ek raasta deta hai: ek time pe ek variable integrate karo.

First principles se DERIVATION (slicing ka argument)

Volume ko slice kyun kiya ja sakta hai? $R$ ke upar $z = f (x, y)$ ke neeche ke solid ki picture socho.

Step 1 — Cross-section se slab volume. $x = x_{0}$ fix karo. $x_{0}$ par vertical plane solid ko ek flat shape mein kaatta hai jiska area hai: $A (x_{0}) = \int_{c}^{d} f (x_{0}, y) d y .$ Yeh step kyun? Us plane par, $x$ frozen hai, isliye curve $y \mapsto f (x_{0}, y)$ ek 1-D region bound karta hai; uska area ek ordinary single integral hai.

Step 2 — Slabs ko jodo. Position $x$ par thickness $d x$ ka ek slab ka volume $A (x) d x$ hai. Total volume: $V = \int_{a}^{b} A (x) d x = \int_{a}^{b} (\int_{c}^{d} f (x, y) d y) d x .$ Yeh step kyun? Yeh exactly $\int (cross-sectional area) d x$ hai — volume-by-slicing formula jo aap single-variable calculus se already jaante ho.

Step 3 — Doosri taraf slice karo. Kuch bhi force nahi kiya tha ki pehle $x$ freeze karein. $y$ freeze karne par $\int_{a}^{b} f (x, y) d x$ area ke slabs milte hain, isliye: $V = \int_{c}^{d} (\int_{a}^{b} f (x, y) d x) d y .$ Yeh step kyun? Usi solid ka volume same hai, isliye dono orders same number dete hain. Yahi equality HAI Fubini's theorem.

Figure — Double integrals over rectangles — Fubini's theorem

Ek special shortcut: separable integrands

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek bread ki loaf hai jiska top weirdly shaped hai. Aap uska total volume chahte ho. Aap ise patli vertical slices mein kaat sakte ho, har slice ke face ka area measure kar sakte ho, aur sab jod sakte ho. Ya aap ise doosri direction mein slice kar sakte ho. Dono taraf se same loaf milti hai, isliye same volume milta hai. Fubini bas kehta hai: "jo bhi taraf easy ho us taraf slice karo, answer same aayega." Double integral bas tiny towers ko add karna hai; iterated integrals unhe row-by-row aur phir column-by-column add karne ka ek saaf tarika hai.

Connections

Riemann sums — double integrals 2-D Riemann limits hain.
Volume by slicing (single-variable) — Fubini ka derivation YEHही idea hai.
Double integrals over general regions — agla step: non-constant inner limits.
Change of order of integration — Fubini use karta hai jab ek order mushkil ho.
Fubini–Tonelli theorem — measure-theoretic generalization.
Triple integrals — teen directions mein slice karo.

Flashcards

Double integral

\iint_{R} f d A

geometrically kya represent karta hai?

Surface

z = f (x, y)

aur

x y

-plane ke beech ka signed volume

R

ke upar (upar ka volume minus neeche ka volume).

Rectangle

R = [a, b] \times [c, d]

ke liye Fubini's theorem state karo.

Agar

f

R

par continuous hai, to

\iint_{R} f d A = \int_{a}^{b} \int_{c}^{d} f d y d x = \int_{c}^{d} \int_{a}^{b} f d x d y

\int_{a}^{b} \int_{c}^{d} f (x, y) d y d x

ke inner integral mein

x

ko kaise treat kiya jaata hai?

Ek constant ki tarah.

Rectangle par dono iteration orders equal kyun hain?

Dono same solid ka volume compute karte hain do alag directions mein slice karke; same solid ⇒ same volume.

\iint_{R} g (x) h (y) d A

ka shortcut?

(\int_{a}^{b} g d x) (\int_{c}^{d} h d y)

, valid kyunki limits constants hain aur variables separate hote hain.

Slicing derivation mein

A (x) = \int_{c}^{d} f (x, y) d y

kya hai?

Fixed

x

par solid ka cross-sectional area.

Fubini kab fail ho sakta hai?

Jab

f

integrable nahi ho (unbounded/non-integrable on

R

); dono orders tab differ kar sakte hain.

\iint_{[0, 2] \times [1, 3]} (x + 2 y) d A = ?

20

Procedure ka mnemonic?

Outer variable ko FREEZE karo, inner 1-D integral ko FILL karo, outer integral mein FILE karo.

Double integrals over rectangles — Fubini's theorem

Double integral HAI kya?

Hum actually compute kaise karte hain? — Iterated integrals

First principles se DERIVATION (slicing ka argument)

Ek special shortcut: separable integrands

Worked Examples

Common Mistakes (Steel-manned)

Connections

Flashcards

Concept Map

Deep Dive